COMPLEX-VALUED ADAPTIVE SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 25

N

The joint pdf for a complex random vector X [ C is extended to the form

N N

f ( x , x ): C

7! R similarly. In the subsequent discussion, we write the expec-
tations with respect to the corresponding joint pdf, pdf of a scalar or vector random
variable as defined here.
Second-order statistics of a complex random vector X are completely defined
through two (auto) covariance matrices: the covariance matrix

C

C XX ¼ E {( XE { X })( XE { X }) H }

that is commonly used, and in addition, the pseudo-covariance [81] matrix—also
called the complementary covariance [101] or the relation matrix [92]—given by

P XX ¼ E {( XE { X })( XE { X }) T } :

Expressions are written similarly for the cross-covariance matrices C XY and P XY
of two complex random vectors X and Y . The properties given in Section 1.2.3 for
complex-to-real mappings can be effectively used to work with covariance matrices
in either the complex- or the double-dimensioned real domain. In the sequel, we
drop the indices used in matrix definitions here when the matrices in question are
clear from the context, and assume that the vectors are zero mean without loss
of generality.
Through their definitions, the covariance matrix is a Hermitian and the pseudo-
covariance matrix is a complex symmetric matrix. As is easily shown, the covariance
matrix is nonnegative definite—and in practice typically positive definite. Hence, the
nonnegative eigenvalues of the covariance matrix can be identified using simple
eigenvalue decomposition. For the pseudo-covariance matrix, however, we need to
use Takagi's factorization [49] to obtain the spectral representation such that

P ¼QDQ T

where Q is a unitary matrix and D ¼ diag fk 1 , k 2 , ... , k N g contains the singular values,
1 k 1 k 2 k N 0, on its diagonal. The values k n are canonical corre-
lations of a given vector and its complex conjugate [100] and are called the circularity
coefficients [33]—though noncircularity coefficients might be the more appropriate
name—since for a second-order circular random vector, which we define next,
these values are all zero.
The vector transformation z [ C

2 N given in (1.13) can be used to
define a single matrix summarizing the second-order properties of a random vector
X , which is called the augmented correlation matrix [92, 101]

N

7! ¯ C [ C

[ X H X T ]

X
X

CP
P C

E { X C X C } ¼ E ¼

and is used in the study of widely linear least mean squares filter which we discuss in
Section 1.4.